Vivo por ella - Andrea Bocelli y Marta Sanchez (HOMENAJE A LA MUSICA)
(Andrea)
Vivo por ella sin saber
si la encontré o me ha encontrado
Ya no recuerdo cómo fue
pero al final me ha conquistado
Vivo por ella que me da
toda mi fuerza de verdad
Vivo por ella y no me pesa
(Marta)
Vivo por ella yo también
no te me pongas tan celoso
Ella entre todas es la más
dulce y caliente como un beso
Ella a mi lado siempre está
para apagar mi soledad
Más que por mi por ella
yo vivo también
(Andrea)
Es la musa que te invita
(Marta)
A tocarla suavecita
(Andrea)
En mi piano a veces triste
la muerte no existe
si ella está aqui
(Marta)
Vivo por ella que me da
todo el afecto que le sale
a veces pega de verdad
(Andrea & Marta)
Pero es un puño que no duele
(Andrea)
Vivo por ella que me da
fuerza, valor y realidad
para sentirme un poco vivo
(Marta)
Cómo duele cuando falta
(Andrea)
Vivo por ella en un hotel
(Marta)
Cómo brilla fuerte y alta
(Andrea)
Vivo por ella en propia piel
(Andrea & Marta)
Si ella canta en mi garganta
mis penas más negras de espanto
(Andrea)
Vivo por ella y nadie más
puede vivir dentro de mi
Ella me da la vida, la vivo
(Andrea & Marta)
Sí está junto a mi
Sí está junto a mi
(Andrea)
Desde un palco o contra un muro
(Marta)
Vivo por ella al límite
(Andrea)
En el trance más oscuro
(Marta)
Vivo por ella íntegra
(Andrea & Marta)
Cada día una conquista
la protagonista
es ella también
(Andrea)
Vivo por ella porque va
dándome simpre la salida
porque la música es así
fiel y sincera de por vida
(Marta)
Vivo por ella que me da
noches de amor y libertad
Si hubiese otra vida, la vivo
por ella también
(Andrea)
Ella se llama música
(Marta)
Yo vivo también
(Andrea)
Vivo por ella créeme
(Marta)
Por ella también
(Andrea & Marta)
Yo vivo per lei
(Andrea)
Yo vivo
(Marta)
Per lei
http://youtu.be/9_i8PVxiN_Q
viernes, 29 de junio de 2012
Bee Gees - Someone belonging to someone 1983
Bee Gees - Someone belonging to someone 1983
I know how I feel
Lonely in the black of night
There can be no love for me now
Nobody listens when the words ain't right
And you get into someone else
It's hard for me to know
Who's lovin' who
And where do I stand
What did I do it all for
There's no one I could love more
I did it for your heart alone
All that I try to be growin' inside to me
I can be strong if you're there
And there is someone belonging to someone
And I got no one belonging to me
I live in a world where the face of an angel
is all that a fool can see
You got the power to find me whenever I'm lost
Where are you ? Who are we?
I believe in time
Can eat away a heart of stone
And baby if I leave you too late
It's just a feather and the bird has flown
It's colder when the fire dies
With all the trouble I'm in if I lose you too
Then what is my life
If I don't wanna go through
Anything without you
Couldn't be life at all
I could be lyin' on
You are the only one
We didn't make it by chance
And there is someone belonging to someone
And I got no one belonging to me
I'm caught in a world on the edge of tomorrow
It's all that a fool can find
I don't belong in the arms of a love that is lost
Nowhere to cry
There must be something we
can say my love
Something except goodbye
Then what is my life
If I don't wanna go through
Anything without you
Couldn't be life at all
I could be lyin' on
You are the only one
We didn't make it by chance
And there is someone belonging to someone
And I got no one belonging to me
And there is someone belonging to someone
And I got no one belonging to me
I know how I feel
Lonely in the black of night
There can be no love for me now
Nobody listens when the words ain't right
And you get into someone else
It's hard for me to know
Who's lovin' who
And where do I stand
What did I do it all for
There's no one I could love more
I did it for your heart alone
All that I try to be growin' inside to me
I can be strong if you're there
And there is someone belonging to someone
And I got no one belonging to me
I live in a world where the face of an angel
is all that a fool can see
You got the power to find me whenever I'm lost
Where are you ? Who are we?
I believe in time
Can eat away a heart of stone
And baby if I leave you too late
It's just a feather and the bird has flown
It's colder when the fire dies
With all the trouble I'm in if I lose you too
Then what is my life
If I don't wanna go through
Anything without you
Couldn't be life at all
I could be lyin' on
You are the only one
We didn't make it by chance
And there is someone belonging to someone
And I got no one belonging to me
I'm caught in a world on the edge of tomorrow
It's all that a fool can find
I don't belong in the arms of a love that is lost
Nowhere to cry
There must be something we
can say my love
Something except goodbye
Then what is my life
If I don't wanna go through
Anything without you
Couldn't be life at all
I could be lyin' on
You are the only one
We didn't make it by chance
And there is someone belonging to someone
And I got no one belonging to me
And there is someone belonging to someone
And I got no one belonging to me
Barbra Streisand / Donna Summer - No More Tears (Enough is Enough)
Barbra Streisand / Donna Summer - No More Tears (Enough is Enough)
Barbra:
It?s raining,
It?s pouring,
My love life is boring me to tears,
After all these years.
Donna:
No sunshine,
No moonlight,
No stardust,
No sign of romance.
We don?t stand a chance.
Barbra:
I always dreamed
I?d find the perfect lover.
But he turned out to be
Like every other man I loved,
I loved.
Both:
Raining (raining)
Pouring (pouring)
There?s nothing left for us here.
And we won?t waste another tear...
Donna:
If you?ve had enough,
Don?t put up
With his stuff,
Don?t you do it.
Barbra:
Now, if you?ve had your fill,
Get the check,
Pay the bill,
You can do it.
Donna:
Tell him to just get out,
Barbra:
Nothing left to talk about.
Donna:
Pack his raincoat, show him out,
Both:
Just look him in the eye and simply shout:
Enough is enough
I can?t go on,
I can?t go on no more no.
Enough is enough
I want him out,
I want him out that door now.
Enough is enough (ooh, owww)
Enough is enough
That?s enough, (ooh yeah)
Donna:
If you?ve reached the end,
Don?t pretend
That it?s right,
When it?s over.
Barbra:
(it?s over)
If the feeling is gone,
Don?t think twice
Just move on,
Get it over. (over, over)
Both:
Tell him to just get out,
Say it clearly,
Spell it out:
Enough is enough is enough
I can?t go on,
I can?t go on no more no.
Enough is enough is enough
I want him out,
I want him out that door now.
Enough is enough (ooh, owww)
Enough is enough
That?s enough (ooh)
Barbra:
I always dreamed I?d find
The perfect lover.
But he turned out to be
Like every other man I loved. (I had no choice from the start)
I loved, (I?ve gotta listen to my heart)
I loved. (tearing us apart)
Both:
Enough is enough is enough
I can?t go on,
I can?t go on no more no.
Enough is enough is enough
I want him out,
I want him out that door now.
Enough is enough (ooh, owww)
Enough is enough
That?s enough (ooh, yeah)
No more tears...
No more tears (no more tears)
No more tears (no more tears)
No more tears (no more tears)
No more tears (no more tears)
I?ve had it, (no more tears)
You?ve had it,
She?s had it. (no more tears)
He?s had it. (no more tears)
She?s had it.
He?s had it.
I?ve had it. (no more tears)
I?ve had it.
He?s had it.
She?s had it.
Donna:
I always dreamed
I?d find the perfect lover
Barbra:
But he turned out to be
Like every other man I loved
Both:
I had no choice from the start
Listen to me, I?ve gotta listen to my heart
Tearing us apart...
Enough is enough is enough
I can?t go on,
I can?t go on no more no.
Enough is enough is enough
I want him out,
I want him out that door now.
Goodbye mister,
Goodbye.
Goodbye mister.
Goodbye sugar.
It?s raining,
It?s pouring,
There?s nothing left for us here.
And we won?t waste another tear... (no more tears)
It?s enough it?s enough it?s enough it?s enough
It?s enough it?s enough it?s enough it?s enough
[guitar]
Enough is enough is enough is enough is enough is enough is
enough
Enough is enough is enough is enough is enough is enough is
enough
[guitar]
Enough is enough is enough is enough
[guitar]
Enough is enough is enough is enough
[guitar]
Enough is enough is enough is enough is enough is enough is
enough
[guitar]
Enough is enough is enough is enough
[guitar]
Enough is enough!
Donna:
I always dreamed
I?d find the perfect lover
Barbra:
But he turned out to be
Like every other man I loved.
Both:
I had no choice from the start.
I?ve gotta listen to my heart, (listen to me, I?ve gotta listen
to my
Heart )
Tearing us apart.
Enough is enough is enough.
I can?t go on,
I can?t go on no more no.
Enough is enough is enough.
I want him out,
I want him out that door now.
Goodbye mister,
Goodbye.
Goodbye mister.
Goodbye sugar.
It?s raining,
It?s pouring.
There?s nothing left for us here,
And we won?t waste another tear... (no more tears)
It?s enough it?s enough it?s enough it?s enough
It?s enough it?s enough it?s enough it?s enough!
Barbra:
It?s raining,
It?s pouring,
My love life is boring me to tears,
After all these years.
Donna:
No sunshine,
No moonlight,
No stardust,
No sign of romance.
We don?t stand a chance.
Barbra:
I always dreamed
I?d find the perfect lover.
But he turned out to be
Like every other man I loved,
I loved.
Both:
Raining (raining)
Pouring (pouring)
There?s nothing left for us here.
And we won?t waste another tear...
Donna:
If you?ve had enough,
Don?t put up
With his stuff,
Don?t you do it.
Barbra:
Now, if you?ve had your fill,
Get the check,
Pay the bill,
You can do it.
Donna:
Tell him to just get out,
Barbra:
Nothing left to talk about.
Donna:
Pack his raincoat, show him out,
Both:
Just look him in the eye and simply shout:
Enough is enough
I can?t go on,
I can?t go on no more no.
Enough is enough
I want him out,
I want him out that door now.
Enough is enough (ooh, owww)
Enough is enough
That?s enough, (ooh yeah)
Donna:
If you?ve reached the end,
Don?t pretend
That it?s right,
When it?s over.
Barbra:
(it?s over)
If the feeling is gone,
Don?t think twice
Just move on,
Get it over. (over, over)
Both:
Tell him to just get out,
Say it clearly,
Spell it out:
Enough is enough is enough
I can?t go on,
I can?t go on no more no.
Enough is enough is enough
I want him out,
I want him out that door now.
Enough is enough (ooh, owww)
Enough is enough
That?s enough (ooh)
Barbra:
I always dreamed I?d find
The perfect lover.
But he turned out to be
Like every other man I loved. (I had no choice from the start)
I loved, (I?ve gotta listen to my heart)
I loved. (tearing us apart)
Both:
Enough is enough is enough
I can?t go on,
I can?t go on no more no.
Enough is enough is enough
I want him out,
I want him out that door now.
Enough is enough (ooh, owww)
Enough is enough
That?s enough (ooh, yeah)
No more tears...
No more tears (no more tears)
No more tears (no more tears)
No more tears (no more tears)
No more tears (no more tears)
I?ve had it, (no more tears)
You?ve had it,
She?s had it. (no more tears)
He?s had it. (no more tears)
She?s had it.
He?s had it.
I?ve had it. (no more tears)
I?ve had it.
He?s had it.
She?s had it.
Donna:
I always dreamed
I?d find the perfect lover
Barbra:
But he turned out to be
Like every other man I loved
Both:
I had no choice from the start
Listen to me, I?ve gotta listen to my heart
Tearing us apart...
Enough is enough is enough
I can?t go on,
I can?t go on no more no.
Enough is enough is enough
I want him out,
I want him out that door now.
Goodbye mister,
Goodbye.
Goodbye mister.
Goodbye sugar.
It?s raining,
It?s pouring,
There?s nothing left for us here.
And we won?t waste another tear... (no more tears)
It?s enough it?s enough it?s enough it?s enough
It?s enough it?s enough it?s enough it?s enough
[guitar]
Enough is enough is enough is enough is enough is enough is
enough
Enough is enough is enough is enough is enough is enough is
enough
[guitar]
Enough is enough is enough is enough
[guitar]
Enough is enough is enough is enough
[guitar]
Enough is enough is enough is enough is enough is enough is
enough
[guitar]
Enough is enough is enough is enough
[guitar]
Enough is enough!
Donna:
I always dreamed
I?d find the perfect lover
Barbra:
But he turned out to be
Like every other man I loved.
Both:
I had no choice from the start.
I?ve gotta listen to my heart, (listen to me, I?ve gotta listen
to my
Heart )
Tearing us apart.
Enough is enough is enough.
I can?t go on,
I can?t go on no more no.
Enough is enough is enough.
I want him out,
I want him out that door now.
Goodbye mister,
Goodbye.
Goodbye mister.
Goodbye sugar.
It?s raining,
It?s pouring.
There?s nothing left for us here,
And we won?t waste another tear... (no more tears)
It?s enough it?s enough it?s enough it?s enough
It?s enough it?s enough it?s enough it?s enough!
Sarah Brightman & Andrea Bocelli - Time to Say Goodbye
Sarah Brightman & Andrea Bocelli - Time to Say Goodbye
Sarah:
Quando sono sola
sogno all'orizzonte
e mancan le parole,
si lo so che non c'e luce
in una stanza quando manca il sole,
se non ci sei tu con me, con me.
Su le finestre
mostra a tutti il mio cuore
che hai accesso,
chiudi dentro me
la luce che
hai incontrato per strada.
Ref:
Time to say goodbye.
Paesi che non ho mai
veduto e vissuto con te,
adesso si li vivro.
Con te partiro
su navi per mari
che, io lo so,
no, no, non esistono piu,
it's time to say goodbye.
Andrea:
Quando sei lontana
sogno all'orizzonte
e mancan le parole,
e io si lo so
che sei con me, con me,
tu mia luna tu sei qui con me,
mio sole tu sei qui con me,
con me, con me, con me.
Ref:
Time to say goodbye.
Paesi che non ho mai
veduto e vissuto con te,
adesso si li vivro.
Con te partiro
su navi per mari
che, io lo so,
no, no, non esistono piu.
Both:
con te io li rivivro.
Con te partiro
su navi per mari
che, io lo so,
no, no, non esistono piu,
con te io li rivivro.
Con te partiro
Io con te.
When I'm alone I dream of the horizon and words fail me.
There is no light in a room where there is no sun
and there is no sun if you're not here with me, with me.
From every window unfurls my heart the heart that you have won.
Into me you've poured the light,
the light that you found by the side of the road.
Time to say goodbye.
Places that I've never seen or experienced with you.
Now I shall, I'll sail with you upon ships across the seas,
seas that exist no more,
it's time to say goodbye.
Andrea Bocelli
When you're far away I dream of the horizon and words fail me.
And of course I know that you're with me, with me.
You, my moon, you are with me.
My sun, you're here with me with me, with me, with me.
Time to say goodbye.
Places that I've never seen or experienced with you.
Now I shall, I'll sail with you upon ships across the seas,
seas that exist no more,
Both
I'll revive them with you.
I'll go with you upon ships across the seas,
seas that exist no more,
I'll revive them with you.
I'll go with you.
You and me.
http://youtu.be/Nl9WMIPzd6w
Sarah:
Quando sono sola
sogno all'orizzonte
e mancan le parole,
si lo so che non c'e luce
in una stanza quando manca il sole,
se non ci sei tu con me, con me.
Su le finestre
mostra a tutti il mio cuore
che hai accesso,
chiudi dentro me
la luce che
hai incontrato per strada.
Ref:
Time to say goodbye.
Paesi che non ho mai
veduto e vissuto con te,
adesso si li vivro.
Con te partiro
su navi per mari
che, io lo so,
no, no, non esistono piu,
it's time to say goodbye.
Andrea:
Quando sei lontana
sogno all'orizzonte
e mancan le parole,
e io si lo so
che sei con me, con me,
tu mia luna tu sei qui con me,
mio sole tu sei qui con me,
con me, con me, con me.
Ref:
Time to say goodbye.
Paesi che non ho mai
veduto e vissuto con te,
adesso si li vivro.
Con te partiro
su navi per mari
che, io lo so,
no, no, non esistono piu.
Both:
con te io li rivivro.
Con te partiro
su navi per mari
che, io lo so,
no, no, non esistono piu,
con te io li rivivro.
Con te partiro
Io con te.
When I'm alone I dream of the horizon and words fail me.
There is no light in a room where there is no sun
and there is no sun if you're not here with me, with me.
From every window unfurls my heart the heart that you have won.
Into me you've poured the light,
the light that you found by the side of the road.
Time to say goodbye.
Places that I've never seen or experienced with you.
Now I shall, I'll sail with you upon ships across the seas,
seas that exist no more,
it's time to say goodbye.
Andrea Bocelli
When you're far away I dream of the horizon and words fail me.
And of course I know that you're with me, with me.
You, my moon, you are with me.
My sun, you're here with me with me, with me, with me.
Time to say goodbye.
Places that I've never seen or experienced with you.
Now I shall, I'll sail with you upon ships across the seas,
seas that exist no more,
Both
I'll revive them with you.
I'll go with you upon ships across the seas,
seas that exist no more,
I'll revive them with you.
I'll go with you.
You and me.
http://youtu.be/Nl9WMIPzd6w
miércoles, 27 de junio de 2012
The Temper Trap - Rabbit Hole
The Temper Trap - Rabbit Hole
I don’t wanna be the only one ’round here
Everything I’m saying’s falling on deaf ears
Come to get you if you make another sound
Make you an example since you call it home
Anyone who listens is gonna get theirs when the water floods down
Into the rabbit hole
Hands up if you think you’re gonna save us now
Before we shoot the messenger for talking too loud
We’ll keep you on the run
We get what we want
Every which way you go they’re gonna hunt you down
Making you believe a lot of sticks and bricks can get you stoned
Cause anyone who listens is gonna get theirs when the water floods down
Into the rabbit hole
You think we gone around
You think we gone around no one
Just feeding the flame
Arresting the fight
And it’s a cold day in hell
It’ll be a cold day in hell
Growing up
Just feeding the flame
Arresting the fight
With fire, with fire
In the rabbit hole
I don’t wanna be the only one ’round here
Everything I’m saying’s falling on deaf ears
Come to get you if you make another sound
Make you an example since you call it home
Anyone who listens is gonna get theirs when the water floods down
Into the rabbit hole
Hands up if you think you’re gonna save us now
Before we shoot the messenger for talking too loud
We’ll keep you on the run
We get what we want
Every which way you go they’re gonna hunt you down
Making you believe a lot of sticks and bricks can get you stoned
Cause anyone who listens is gonna get theirs when the water floods down
Into the rabbit hole
You think we gone around
You think we gone around no one
Just feeding the flame
Arresting the fight
And it’s a cold day in hell
It’ll be a cold day in hell
Growing up
Just feeding the flame
Arresting the fight
With fire, with fire
In the rabbit hole
Scorpions - Rhythm Of Love
Scorpions - Rhythm Of Love
Let's spend the night together
I know you want it too
The magic of the moment
Is what I've got for you
The heartbeat of this night
Is made to lose control
And there is something in your eyes
That's longing for some more
Let us find together
The beat we're looking for
The rhythm of love
Keeps me dancing on the road
The rhythm of love
Got the groove that hits the bone
The rhythm of love
Is the game I'm looking for
The rhythm of love
Is the heartbeat of my soul
Let's reach the top together
One night will never do
An exploding shot of pleasure
Is what I've got for you
Why don't you close your eyes
And let your feeling grow
I make you feel the taste of life
Until your love will flow
Let us find together
The beat we're longing for
The rhythm of love
Keeps me dancing on the road
The rhythm of love
Got the groove that hits the bone
The rhythm of love
Is the game I'm looking for
The rhythm of love
Is the heartbeat of my soul
The rhythm of love
Keeps me dancing on the road
The rhythm of love
Got the groove that hits the bone
The rhythm of love
Is the game I'm looking for
The rhythm of love
Is the heartbeat of my soul
The rhythm of love
Keeps me dancing on the road
The rhythm of love
Got the groove that hits the bone
The rhythm of love
Is the game I'm looking for
The rhythm of love
Is the heartbeat of my soul
Let us find together
The beat we're looking for
Let's spend the night together
I know you want it too
The magic of the moment
Is what I've got for you
The heartbeat of this night
Is made to lose control
And there is something in your eyes
That's longing for some more
Let us find together
The beat we're looking for
The rhythm of love
Keeps me dancing on the road
The rhythm of love
Got the groove that hits the bone
The rhythm of love
Is the game I'm looking for
The rhythm of love
Is the heartbeat of my soul
Let's reach the top together
One night will never do
An exploding shot of pleasure
Is what I've got for you
Why don't you close your eyes
And let your feeling grow
I make you feel the taste of life
Until your love will flow
Let us find together
The beat we're longing for
The rhythm of love
Keeps me dancing on the road
The rhythm of love
Got the groove that hits the bone
The rhythm of love
Is the game I'm looking for
The rhythm of love
Is the heartbeat of my soul
The rhythm of love
Keeps me dancing on the road
The rhythm of love
Got the groove that hits the bone
The rhythm of love
Is the game I'm looking for
The rhythm of love
Is the heartbeat of my soul
The rhythm of love
Keeps me dancing on the road
The rhythm of love
Got the groove that hits the bone
The rhythm of love
Is the game I'm looking for
The rhythm of love
Is the heartbeat of my soul
Let us find together
The beat we're looking for
Scorpions - Believe In Love
Scorpions - Believe In Love
How does it fell babe
To taste sweet revenge
Do you want me
on my knees
How does it feel babe
to let me feel your strength
Don't be cruel,
can't you see
If you don't catch me now
I can't stop falling down
Just one more night and the devil's got my soul
I need your love babe
don't tell me
"no way"
Babe I miss you
I miss you
Baby our love's got what it takes
To give us one more chance to start
once again
Baby our love will find a way
As long as we believe in love
If you don't catch me now
I can't stop falling down
Just one more night and the devil's got my soul
I need your love babe
don't leave me
this way
Ooh I miss you,
I miss you
Baby our love's got what it takes
To give us one more chance to start
once again
Baby our love will find a way
As long as we believe in love
As long as we believe
we believe
Baby our love's got what it takes
To give us one more chance to start
once again
Baby our love will find a way
As long as we believe in love
Baby our love's got what it takes
To give us one more chance to start
once again
Baby our love will find a way
As long as we believe in love
How does it fell babe
To taste sweet revenge
Do you want me
on my knees
How does it feel babe
to let me feel your strength
Don't be cruel,
can't you see
If you don't catch me now
I can't stop falling down
Just one more night and the devil's got my soul
I need your love babe
don't tell me
"no way"
Babe I miss you
I miss you
Baby our love's got what it takes
To give us one more chance to start
once again
Baby our love will find a way
As long as we believe in love
If you don't catch me now
I can't stop falling down
Just one more night and the devil's got my soul
I need your love babe
don't leave me
this way
Ooh I miss you,
I miss you
Baby our love's got what it takes
To give us one more chance to start
once again
Baby our love will find a way
As long as we believe in love
As long as we believe
we believe
Baby our love's got what it takes
To give us one more chance to start
once again
Baby our love will find a way
As long as we believe in love
Baby our love's got what it takes
To give us one more chance to start
once again
Baby our love will find a way
As long as we believe in love
martes, 26 de junio de 2012
jueves, 21 de junio de 2012
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Grafica de una Derivada vs funcion - Equilibria and the Phase Line
http://www.sosmath.com/diffeq/first/phaseline/phaseline.html
Solving a differential equation can be done in three major ways: analytical, qualitative, and numerical. We have seen some examples of differential equations solved through analytical techniques (for example: linear, separable, and Bernoulli equations). Euler's Method (though very primitive) illustrates the use of numerical techniques in solving differential equations. For the qualitative approach, we have defined the slope field of a differential equation and showed how this can be helpful when other techniques fail.
When the differential equation is autonomous, more can be said about the solutions using qualitative techniques.
Consider the autonomous differential equation
where f(y) is a function. Clearly we recognize a separable equation. So we know how to solve it via the analytical technique. First we look for the constant solutions as the roots of the equation
Then to find the nonconstant solutions we separate the variable
and integrate
Note that the integration may be hard if not impossible to carry out. So it may not be possible to use the analytical technique.
In this case, one may try numerical techniques. Unfortunately, numerical techniques fail if f(y) contains a parameter and we need to make conclusions relative to this parameter. For example, recall the logistic equation with harvesting (with a constant rate H)
where y(t) is the population of the given species at time t. In order to find the optimal rate H (which makes hunters happy as well as ecologists worried about preserving the species), numerical techniques may not be the best tool to use. Here we will see how new ideas based on a graphical approach will help us to say something about the solutions. Be aware that these ideas are only valid for autonomous equations.
Equilibria of Autonomous Equations
Consider the autonomous equation
The equilibria or constant solutions of this differential equation are the roots of the equation
Using the existence and uniqueness theorem, these constant solutions will cut the entire plane (where the solutions live) into independent regions. This means, if an initial condition belongs to one of the regions, then the solution satisfying the initial condition will stay in that region all the time.
Example. Consider the logistic equation
Its equilibria are y=0 and y=1. Consider the solution y(t) to the IVP
Since 0 <y(0) < 1, then we must have
In the next picture, we draw few solutions associated to initial conditions in the different regions (0 < y < 1, y < 0, and 1 < y)
Note that in this example, any solution y(t) is increasing if 0 <
y(0) < 1 is satisfied. So one may wonder whether a
similar conclusion is true in general. Consider again the autonomous equation
Assume that
and 
( 
) are equilibria (that is 
and
).
Assume moreover that f(y) has a constant sign on the
interval 
.
Assume that f(y) is positive and consider the
solution to the IVP
where
. We know that we will have 
for every t. Since
we must have
for all t. This
clearly implies that y(t) is always increasing. So the sign of
f(y) helps us determine the behavior of the solutions (whether they are
increasing or decreasing).
Example. Consider the autonomous equation
where the graph of f(y) is given below
From the graph, we can determine the equilibria. They are y=0, y=1, and
y=2. Using the discussion above, we obtain the following information:
Remark. Recall that if the function h(t) is increasing or decreasing, then the following hold :
, and h(t) is a "nice function", then we must have
The next image illustrates this conclusion better than words can do.
Example. Consider the solution to the initial value problem
where the graph of f(y) is given below
Discuss the limit of y(t) when .
Answer. Using the above results, we know that for the given initial condition, y(t) is decreasing and for every t we have
The above remark implies that
exist and we have
Since y(t) is not a constant solution (because y(0)=0.5), then we have
On the other hand, since
and y is solution to the autonomous equation y'=f(y), then the only possibilities for both limits are the equilibria 0,1, and 2. Clearly if we put everything together, we conclude that
Summary. Let us write down some general steps to follow when dealing with the autonomous equation
Example. Draw some solutions for the equation
Answer. Note that this equation models the logistic growth with threshold. The equilibria or constant solutions are given by
or y=0, y=2, and y=5. The graph of the function f(y) = .2y(5-y)(y-2) is given in the next picture
So we have:
Phase Line
Let us reconsider the above example. Let us focus our attention on the solutions which satisfy the initial condition
and 
.
We already know that
It is somehow amazing that this conclusion is valid regardless whether is close
to 5 or close to 2. In this case we say that the
equilibrium point 5 is attractive from below. In fact, in this
example, this equilibrium point is attractive from below and above.
On the other hand, the solutions get away from the equilibrium point 2
from below and above. We will say that the equilibrium point 2 is
repelling. Note that there is no general agreement on the words used to describe these phenomena.
Definition. Classification of Equilibrium Points.
Consider the autonomous equation
Assume that is an equilibrium point (that is ).
as source, sink, or node.
Answer. The equilibrium points are 0, 2, and 5. Using the above results, we see that 0 and 5 are sinks while 2 is a source.
Example. Find and classify the equilibrium points of the equation
as source, sink, or node.
Answer. It is easy to see that the only equilibrium point is 0. Since
is always positive, then the solutions will always
be increasing. Using previous results, we conclude:
There is a very nice way to put all this information together.
Indeed, draw a vertical line (where the variable describing it
is y) and start by marking the equilibrium points of the equation
on the y-axis.
Then using the sign of f(y), we draw arrows pointing upward in a region where f(y) is positive, and downward in a region where f(y) is negative. This vertical line is called the phase line of the equation.
An equilibrium point is a sink, if the arrows on both sides point towards the equilibrium point, and it is a source, if both arrows point away from it.
Example. Draw the phase line of the equations
and
Answer. We will use our previous knowledge to get the two phase lines
and
You can readily see, that an equilibrium point is a sink, if the arrows on both sides point towards the equilibrium point, and that it is a source, if both arrows point away from it.
[Differential Equations]
[Slope Field]
[Bifurcations]
[Geometry]
[Algebra]
[Trigonometry ]
[Calculus]
[Complex Variables]
[Matrix Algebra]
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Equilibria and the Phase Line
Solving a differential equation can be done in three major ways: analytical, qualitative, and numerical. We have seen some examples of differential equations solved through analytical techniques (for example: linear, separable, and Bernoulli equations). Euler's Method (though very primitive) illustrates the use of numerical techniques in solving differential equations. For the qualitative approach, we have defined the slope field of a differential equation and showed how this can be helpful when other techniques fail.
When the differential equation is autonomous, more can be said about the solutions using qualitative techniques.
Consider the autonomous differential equation
where f(y) is a function. Clearly we recognize a separable equation. So we know how to solve it via the analytical technique. First we look for the constant solutions as the roots of the equation
Then to find the nonconstant solutions we separate the variable
and integrate
Note that the integration may be hard if not impossible to carry out. So it may not be possible to use the analytical technique.
In this case, one may try numerical techniques. Unfortunately, numerical techniques fail if f(y) contains a parameter and we need to make conclusions relative to this parameter. For example, recall the logistic equation with harvesting (with a constant rate H)
where y(t) is the population of the given species at time t. In order to find the optimal rate H (which makes hunters happy as well as ecologists worried about preserving the species), numerical techniques may not be the best tool to use. Here we will see how new ideas based on a graphical approach will help us to say something about the solutions. Be aware that these ideas are only valid for autonomous equations.
Equilibria of Autonomous Equations
Consider the autonomous equation
The equilibria or constant solutions of this differential equation are the roots of the equation
Using the existence and uniqueness theorem, these constant solutions will cut the entire plane (where the solutions live) into independent regions. This means, if an initial condition belongs to one of the regions, then the solution satisfying the initial condition will stay in that region all the time.
Example. Consider the logistic equation
Its equilibria are y=0 and y=1. Consider the solution y(t) to the IVP
Since 0 <y(0) < 1, then we must have
In the next picture, we draw few solutions associated to initial conditions in the different regions (0 < y < 1, y < 0, and 1 < y)
|
Assume that
and ( ) are equilibria (that is and
).
Assume moreover that f(y) has a constant sign on the
interval .
Assume that f(y) is positive and consider the
solution to the IVP
where
. We know that we will have
for every t. Since
we must have
for all t. This
clearly implies that y(t) is always increasing. So the sign of
f(y) helps us determine the behavior of the solutions (whether they are
increasing or decreasing). Example. Consider the autonomous equation
where the graph of f(y) is given below
|
-
- if y(t) is a solution satisfying y(0) < 0, then y(t) is always increasing;
-
- if y(t) is a solution satisfying 0<y(0) < 1, then y(t) is always decreasing;
-
- if y(t) is a solution satisfying 1 <y(0) < 2, then y(t) is always increasing;
-
- if y(t) is a solution satisfying 2 <y(0), then y(t) is always decreasing.
|
-
- the limit of h(t) when
exists (or is a number) if and only if the function is bounded above; otherwise we have
-
- the limit of h(t) when
exists (or is a number) if and only if the function is bounded below; otherwise we have
, and h(t) is a "nice function", then we must have
The next image illustrates this conclusion better than words can do.
|
where the graph of f(y) is given below
|
.Answer. Using the above results, we know that for the given initial condition, y(t) is decreasing and for every t we have
The above remark implies that
exist and we have
Since y(t) is not a constant solution (because y(0)=0.5), then we have
On the other hand, since
and y is solution to the autonomous equation y'=f(y), then the only possibilities for both limits are the equilibria 0,1, and 2. Clearly if we put everything together, we conclude that
|
- Step 1. Find the constant solutions or equilibria through the algebraic equation
Draw the constant solutions. Note that here we are drawing y versus t. - Step 2. Find the sign of the function f(y) through its graph (versus y).
- Step 3. In the region between any two equilibria, the solutions will be all increasing (if f(y) is positive) or decreasing (if f(y) is negative).
- Step 4. If a solution is increasing and bounded above by a constant solution y = L, then the limit of the solution when is the number L. Otherwise the limit is
.
If this solution is bounded below by a constant solution y = l, then the limit of the solution when is the number l. Otherwise the limit is 
.
- Step 5. If a solution is decreasing and bounded below by a constant solution y = L, then the limit of the solution when is the number L. Otherwise the limit is .
If this solution is bounded above by a constant solution y = l, then the limit of the solution when is the number l. Otherwise the limit is .
- Step 6. Draw some solutions.
Example. Draw some solutions for the equation
Answer. Note that this equation models the logistic growth with threshold. The equilibria or constant solutions are given by
or y=0, y=2, and y=5. The graph of the function f(y) = .2y(5-y)(y-2) is given in the next picture
|
-
- the solution which satisfies the initial condition
, with
, is increasing. Moreover we have
-
- the solution which satisfies the initial condition
, with
, is decreasing. Moreover we have
-
- the solution which satisfies the initial condition
, with , is increasing. Moreover we have
-
- the solution which satisfies the initial condition
, with
, is decreasing. Moreover we have
|
Let us reconsider the above example. Let us focus our attention on the solutions which satisfy the initial condition
and .
We already know that
It is somehow amazing that this conclusion is valid regardless whether
is close
to 5 or close to 2. In this case we say that the
equilibrium point 5 is attractive from below. In fact, in this
example, this equilibrium point is attractive from below and above.
On the other hand, the solutions get away from the equilibrium point 2
from below and above. We will say that the equilibrium point 2 is
repelling. Note that there is no general agreement on the words used to describe these phenomena.Definition. Classification of Equilibrium Points.
Consider the autonomous equation
Assume that
is an equilibrium point (that is ).
-
- The equilibrium point is called a sink
if and only if any solution y(t) to the autonomous equation such
that y(0) is close to , then we have
-
- The equilibrium point is called a source
if and only if any solution y(t) to the autonomous equation such
that y(0) is (or not) close to , then y(t) will move away from
when .
-
- The equilibrium point is called a node
if and only if it is neither a source nor a sink.
as source, sink, or node.
Answer. The equilibrium points are 0, 2, and 5. Using the above results, we see that 0 and 5 are sinks while 2 is a source.
Example. Find and classify the equilibrium points of the equation
as source, sink, or node.
Answer. It is easy to see that the only equilibrium point is 0. Since
is always positive, then the solutions will always
be increasing. Using previous results, we conclude:
-
- if y(t) is a solution satisfying y(0) < 0, then
we have
-
- if y(t) is a solution satisfying y(0) > 0, then
we have
|
on the y-axis.
Then using the sign of f(y), we draw arrows pointing upward in a region where f(y) is positive, and downward in a region where f(y) is negative. This vertical line is called the phase line of the equation.
An equilibrium point is a sink, if the arrows on both sides point towards the equilibrium point, and it is a source, if both arrows point away from it.
Example. Draw the phase line of the equations
and
Answer. We will use our previous knowledge to get the two phase lines
|
|
You can readily see, that an equilibrium point is a sink, if the arrows on both sides point towards the equilibrium point, and that it is a source, if both arrows point away from it.
S.O.S. MATHematics home page Last Update 6-22-98
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